函数项级数中的一致收敛与极限运算交换次序

On the Uniform Convergence and Exchanging Orders of Limit Operations in Functional Series

基于逐项取极限定理,从新的视角对函数项级数的分析性质涉及的极限运算交换次序问题及其条件进行了讨论,解释"一致收敛"为何在函数项级数的极限运算交换中起作用以及在哪一环节中发挥作用.本文讨论有助于在教学中更清晰地理解函数项级数中的一致收敛与极限运算交换次序,深入理解和思考数学的理论性。

From a new point of view, this paper discusses the analytic properties of exchanging orders of limit operations in function series and the related conditions. It explains why and where uniform convergence plays critical roles in exchanging orders of limit operations in function series. The analysis in this paper will help the students better understand the relationship between the uniform convergence and the exchanging orders of limit operations in function series, and thus more deeply understand the theories of mathematics.